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Problem 1
• The spring is un-stretched when θ = 30
o
. At any
position of the pendulum, the spring remains
horizontal. If the spring constant is k = 50 N/m, at
what position will the system be in equilibrium.
( Soln: No need to solve this problem. It involves
Newton-Raphson like method for actual
solution)
Problem 2
• If the springs are un-
stretched when θ = θ
o
find the angle θ when
the weight W is applied
on the system. Use the
method of minimum
potential energy.
θ = 0 ; cos(θ) = cos(θ
W
(k
+ k
)a
Problem 3
• Two bars are attached to a single spring of
constant k that is un-stretched when the bars are
vertical. Determine the range of values of P for
which the equilibrium of the system is stable in the
position shown.
P ≤
ka
P ≤
ka
4 L
( stability both for parts (a) and (b))
Problem 4
• The horizontal bar AD is attached to two springs of constant
k and is in equilibrium in the position shown. Determine the
range of values of the magnitude P of the two equal and
opposite horizontal forces P and –P for which the equilibrium
position is stable if (a) AB = CD, (b) if AB = 2CD.
o
, stable; θ = 207
o
unstable
Problem 6
• Determine the equilibrium values of θ and the
stability of equilibrium at each point for the
unbalanced wheel on the 10
o
incline. Static friction
is sufficient to prevent slipping. The mass center is
at G.
Extra Problem 1
• The uniform disk of radius R and mass m rolls without
slipping on the fixed cylinder surface of radius 2R. Fastened
to the disk is a lead cylinder also of mass m with its center
located a distance b from the center O of the disk. Determine
the minimum value of b for which the disk will remain in
stable equilibrium on the cylindrical surface.
b
min
R
Extra Problem 3
- One of the critical requirements in the design of an artificial leg for
an amputee is to prevent the knee joint from buckling under load
when the leg is straight. As a first approximation, simulate the
artificial leg by the two light links with a torsion spring at their
common joint. The spring develops a torque M = K β, which Is
proportional to the angle of the bend β at the joint. Determine the
minimum value of K which will ensure the stability of the knee joint
for β = 0.
K
min
mgl
Extra Problem 4
• The horizontal bar BEH is pinned to collar E and to
vertical bars AC and GI. The collar can slide freely
on bar DF. Determine the range of values of Q for
which the equilibrium of the system is stable in the
position shown when a = 480mm, b = 400mm, and
P = 600N.
Q > 432 N
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1798
PROBLEM 10.
A slender rod AB , of weight W , is attached to two blocks A
and B that can move freely in the guides shown. Knowing that
the spring is unstretched when y =0,determine the value of
y corresponding to equilibrium when W =80 N, l =500 mm,
and k =600 N/m.
SOLUTION
Deflection of spring = s , where
2 2
2 2
= + −
=
−
s l y l
ds y
dy l y
Potential energy:
( )
2
2 2
2 2
2 2
y
V ks W
dV ds
ks W
dy dy
dV y
k l y l W
dy l y
l
k y W
l y
Equilibrium
2 2
= ¨^ − ¸ =
dV l W
y
dy k l y
Now W = 80 N, l = 0.500 m, and k =600 N/m
Then
2 2
0.500 m 1 (80 N)
2 (600 N/m) (0.500)
y
y
or
2
y
y
Solving numerically, y =0.357 m y =357 mmW
